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2.9.4. Photosynthesis¶
Photosynthesis in C3 plants is based on the model of Farquhar et al. (1980). Photosynthesis in C4 plants is based on the model of Collatz et al. (1992). Bonan et al. (2011) describe the implementation, modified here. In its simplest form, leaf net photosynthesis after accounting for respiration (\(R_{d}\) ) is
(2.9.2)¶\[A_{n} =\min \left(A_{c} ,A_{j} ,A_{p} \right)-R_{d} .\]
The RuBP carboxylase (Rubisco) limited rate of carboxylation \(A_{c}\) (\(\mu\) mol CO2 m-2 s-1) is
(2.9.3)¶\[\begin{split}A_{c} =\left\{\begin{array}{l} {\frac{V_{c\max } \left(c_{i} -\Gamma _{*} \right)}{c_{i} +K_{c} \left(1+{o_{i} \mathord{\left/ {\vphantom {o_{i} K_{o} }} \right.} K_{o} } \right)} \qquad {\rm for\; C}_{{\rm 3}} {\rm \; plants}} \\ {V_{c\max } \qquad \qquad \qquad {\rm for\; C}_{{\rm 4}} {\rm \; plants}} \end{array}\right\}\qquad \qquad c_{i} -\Gamma _{*} \ge 0.\end{split}\]
The maximum rate of carboxylation allowed by the capacity to regenerate RuBP (i.e., the light-limited rate) \(A_{j}\) (\(\mu\) mol CO2 m-2 s-1) is
(2.9.4)¶\[\begin{split}A_{j} =\left\{\begin{array}{l} {\frac{J_{x}\left(c_{i} -\Gamma _{*} \right)}{4c_{i} +8\Gamma _{*} } \qquad \qquad {\rm for\; C}_{{\rm 3}} {\rm \; plants}} \\ {\alpha (4.6\phi )\qquad \qquad {\rm for\; C}_{{\rm 4}} {\rm \; plants}} \end{array}\right\}\qquad \qquad c_{i} -\Gamma _{*} \ge 0.\end{split}\]
The product-limited rate of carboxylation for C3 plants and the PEP carboxylase-limited rate of carboxylation for C4 plants \(A_{p}\) (\(\mu\) mol CO2 m-2 s-1) is
(2.9.5)¶\[\begin{split}A_{p} =\left\{\begin{array}{l} {3T_{p\qquad } \qquad \qquad {\rm for\; C}_{{\rm 3}} {\rm \; plants}} \\ {k_{p} \frac{c_{i} }{P_{atm} } \qquad \qquad \qquad {\rm for\; C}_{{\rm 4}} {\rm \; plants}} \end{array}\right\}.\end{split}\]
In these equations, \(c_{i}\) is the internal leaf CO2 partial pressure (Pa) and \(o_{i} =0.20P_{atm}\) is the O2 partial pressure (Pa). \(K_{c}\) and \(K_{o}\) are the Michaelis-Menten constants (Pa) for CO2 and O2. \(\Gamma _{*}\) (Pa) is the CO2 compensation point. \(V_{c\max }\) is the maximum rate of carboxylation (µmol m-2 s-1, Chapter 2.10) and \(J_{x}\) is the electron transport rate (µmol m-2 s-1). \(T_{p}\) is the triose phosphate utilization rate (µmol m-2 s-1), taken as \(T_{p} =0.167V_{c\max }\) so that \(A_{p} =0.5V_{c\max }\) for C3 plants (as in Collatz et al. 1992). For C4 plants, the light-limited rate \(A_{j}\) varies with \(\phi\) in relation to the quantum efficiency (\(\alpha =0.05\) mol CO2 mol-1 photon). \(\phi\) is the absorbed photosynthetically active radiation (W m-2) (section 2.4.1), which is converted to photosynthetic photon flux assuming 4.6 \(\mu\) mol photons per joule. \(k_{p}\) is the initial slope of C4 CO2 response curve.
For C3 plants, the electron transport rate depends on the photosynthetically active radiation absorbed by the leaf. A common expression is the smaller of the two roots of the equation
(2.9.6)¶\[\Theta _{PSII} J_{x}^{2} -\left(I_{PSII} +J_{\max } \right)J_{x}+I_{PSII} J_{\max } =0\]
where \(J_{\max }\) is the maximum potential rate of electron transport (\(\mu\)mol m-2 s-1, Chapter 2.10), \(I_{PSII}\) is the light utilized in electron transport by photosystem II (µmol m-2 s-1), and \(\Theta _{PSII}\) is a curvature parameter. For a given amount of photosynthetically active radiation absorbed by a leaf (\(\phi\), W m-2), converted to photosynthetic photon flux density with 4.6 \(\mu\)mol J-1, the light utilized in electron transport is
(2.9.7)¶\[I_{PSII} =0.5\Phi _{PSII} (4.6\phi )\]
where \(\Phi _{PSII}\) is the quantum yield of photosystem II, and the term 0.5 arises because one photon is absorbed by each of the two photosystems to move one electron. Parameter values are \(\Theta _{PSII}\) = 0.7 and \(\Phi _{PSII}\) = 0.85. In calculating \(A_{j}\) (for both C3 and C4 plants), \(\phi =\phi ^{sun}\) for sunlit leaves and \(\phi =\phi ^{sha}\) for shaded leaves.
The model uses co-limitation as described by Collatz et al. (1991, 1992). The actual gross photosynthesis rate, \(A\), is given by the smaller root of the equations
(2.9.8)¶\[\begin{split}\begin{array}{rcl} {\Theta _{cj} A_{i}^{2} -\left(A_{c} +A_{j} \right)A_{i} +A_{c} A_{j} } & {=} & {0} \\ {\Theta _{ip} A^{2} -\left(A_{i} +A_{p} \right)A+A_{i} A_{p} } & {=} & {0} \end{array} .\end{split}\]
Values are \(\Theta _{cj} =0.98\) and \(\Theta _{ip} =0.95\) for C3 plants; and \(\Theta _{cj} =0.80\)and \(\Theta _{ip} =0.95\) for C4 plants. \(A_{i}\) is the intermediate co-limited photosynthesis. \(A_{n} =A-R_{d}\).
The parameters \(K_{c}\), \(K_{o}\), and \(\Gamma\) depend on temperature. Values at 25 °C are \(K_{c25} ={\rm 4}0{\rm 4}.{\rm 9}\times 10^{-6} P_{atm}\), \(K_{o25} =278.4\times 10^{-3} P_{atm}\), and \(\Gamma _{25} {\rm =42}.75\times 10^{-6} P_{atm}\). \(V_{c\max }\), \(J_{\max }\), \(T_{p}\), \(k_{p}\), and \(R_{d}\) also vary with temperature.
\(J_{\max 25}\) at 25 oC: is calculated by the LUNA model (Chapter 2.10)
Parameter values at 25 oC are calculated from \(V_{c\max }\) at 25 oC:, including: \(T_{p25} =0.167V_{c\max 25}\), and \(R_{d25} =0.015V_{c\max 25}\) (C3) and \(R_{d25} =0.025V_{c\max 25}\) (C4).
For C4 plants, \(k_{p25} =20000\; V_{c\max 25}\).
However, when the biogeochemistry is active (the default mode), \(R_{d25}\) is calculated from leaf nitrogen as described in (Chapter 2.17)
The parameters \(V_{c\max 25}\), \(J_{\max 25}\), \(T_{p25}\), \(k_{p25}\), and \(R_{d25}\) are scaled over the canopy for sunlit and shaded leaves (section 2.9.5). In C3 plants, these are adjusted for leaf temperature, \(T_{v}\) (K), as:
(2.9.9)¶\[\begin{split}\begin{array}{rcl} {V_{c\max } } & {=} & {V_{c\max 25} \; f\left(T_{v} \right)f_{H} \left(T_{v} \right)} \\ {J_{\max } } & {=} & {J_{\max 25} \; f\left(T_{v} \right)f_{H} \left(T_{v} \right)} \\ {T_{p} } & {=} & {T_{p25} \; f\left(T_{v} \right)f_{H} \left(T_{v} \right)} \\ {R_{d} } & {=} & {R_{d25} \; f\left(T_{v} \right)f_{H} \left(T_{v} \right)} \\ {K_{c} } & {=} & {K_{c25} \; f\left(T_{v} \right)} \\ {K_{o} } & {=} & {K_{o25} \; f\left(T_{v} \right)} \\ {\Gamma } & {=} & {\Gamma _{25} \; f\left(T_{v} \right)} \end{array}\end{split}\]
(2.9.10)¶\[f\left(T_{v} \right)=\; \exp \left[\frac{\Delta H_{a} }{298.15\times 0.001R_{gas} } \left(1-\frac{298.15}{T_{v} } \right)\right]\]
and
(2.9.11)¶\[f_{H} \left(T_{v} \right)=\frac{1+\exp \left(\frac{298.15\Delta S-\Delta H_{d} }{298.15\times 0.001R_{gas} } \right)}{1+\exp \left(\frac{\Delta ST_{v} -\Delta H_{d} }{0.001R_{gas} T_{v} } \right)} .\]
Table 2.9.2 lists parameter values for \(\Delta H_{a}\) and \(\Delta H_{d}\). \(\Delta S\) is calculated separately for \(V_{c\max }\) and \(J_{max }\) to allow for temperature acclimation of photosynthesis (see equation (2.9.16)), and \(\Delta S\) is 490 J mol -1 K -1 for \(R_d\) (Bonan et al. 2011, Lombardozzi et al. 2015). Because \(T_{p}\) as implemented here varies with \(V_{c\max }\), \(T_{p}\) uses the same temperature parameters as \(V_{c\max}\). For C4 plants,
(2.9.12)¶\[\begin{split}\begin{array}{l} {V_{c\max } =V_{c\max 25} \left[\frac{Q_{10} ^{(T_{v} -298.15)/10} }{f_{H} \left(T_{v} \right)f_{L} \left(T_{v} \right)} \right]} \\ {f_{H} \left(T_{v} \right)=1+\exp \left[s_{1} \left(T_{v} -s_{2} \right)\right]} \\ {f_{L} \left(T_{v} \right)=1+\exp \left[s_{3} \left(s_{4} -T_{v} \right)\right]} \end{array}\end{split}\]
with \(Q_{10} =2\), \(s_{1} =0.3\)K-1 \(s_{2} =313.15\) K, \(s_{3} =0.2\)K-1, and \(s_{4} =288.15\) K. Additionally,
(2.9.13)¶\[R_{d} =R_{d25} \left\{\frac{Q_{10} ^{(T_{v} -298.15)/10} }{1+\exp \left[s_{5} \left(T_{v} -s_{6} \right)\right]} \right\}\]
with \(Q_{10} =2\), \(s_{5} =1.3\) K-1 and \(s_{6} =328.15\)K, and
(2.9.14)¶\[k_{p} =k_{p25} \, Q_{10} ^{(T_{v} -298.15)/10}\]
with \(Q_{10} =2\).
Table 2.9.2 Temperature dependence parameters for C3 photosynthesis.¶
| Parameter
| \(\Delta H_{a}\) (J mol-1)
| \(\Delta H_{d}\) (J mol-1)
| \(V_{c\max }\) |
| 72000
| 200000
| | \(J_{\max }\)
| 50000
| 200000
| | \(T_{p}\)
| 72000
| 200000
| | \(R_{d}\)
| 46390
| 150650
| | \(K_{c}\)
| 79430
| –
| | \(K_{o}\)
| 36380
| –
| | \(\Gamma _{*}\)
| 37830
| –
|
In the model, acclimation is implemented as in Kattge and Knorr (2007). In this parameterization, \(V_{c\max }\) and \(J_{\max }\) vary with the plant growth temperature. This is achieved by allowing \(\Delta S\)to vary with growth temperature according to
(2.9.15)¶\[\begin{split}\begin{array}{l} {\Delta S=668.39-1.07(T_{10} -T_{f} )\qquad \qquad {\rm for\; }V_{c\max } } \\ {\Delta S=659.70-0.75(T_{10} -T_{f} )\qquad \qquad {\rm for\; }J_{\max } } \end{array}\end{split}\]
The effect is to cause the temperature optimum of \(V_{c\max }\) and \(J_{\max }\) to increase with warmer temperatures. Additionally, the ratio \(J_{\max 25} /V_{c\max 25}\) at 25 °C decreases with growth temperature as
(2.9.16)¶\[J_{\max 25} /V_{c\max 25} =2.59-0.035(T_{10} -T_{f} ).\]
In these acclimation functions, \(T_{10}\) is the 10-day mean air temperature (K) and \(T_{f}\) is the freezing point of water (K). For lack of data, \(T_{p}\) acclimates similar to \(V_{c\max }\). Acclimation is restricted over the temperature range \(T_{10} -T_{f} \ge\) 11°C and \(T_{10} -T_{f} \le\) 35°C.